Coupling of two competitive systems via density dependent migration

Coupling of two Lotka–Volterra type competition systems with density-dependent migration was surveyed. We assumed that species x and y are each exclusively superior in subhabitat 1 and subhabitat 2, respectively, and that population densities that exert intra-and interspecific competitive effects also impose pressures for migration of individuals from a subhabitat. If the two species are, respectively, abundant in the subhabitats in which either species is competitively superior, and the migration has a mixing effect, then, it would be intuitively expected that, as potential migration rates increase, the two species are mixed well and coexist in the whole habitat. An analysis of this competitive situation using our model under the assumption of linear diffusion predicted that, even though weak mixing maintains coexistence in the whole habitat, strong mixing collapses coexistence and leads to the exclusion of one species. The assumption that migrations occur due to self- and cross-population pressures provides different predictions: (i) weak dominance and strong mixing destabilize the coexistence state and lead to a monopolizing equilibrium of either species (bi-stability of monopolizing equiliblia); (ii) conspicuous weakness of the inferior species makes the mixing equilibrium stable, regardless of the potential migration rate; and (iii) tri-stability exists in between situations (i) and (ii). In the third case, the attainable state is the mixing equilibrium or either of the monopolizing equilibria, depending on the initial state. Migration mechanisms with self- and cross-population pressures tends to mediate spatial segregation and makes coexistence possible, even with strong mixing.